Subgroups — C₂²×#500 #60654

Order 256.

1199 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×#500	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#107	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈	—	2	2	no	—
6	C₂³×C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈	—	2	12	no	—
8	C₂×C₄⋊Q₈	C₂²	1	24	yes	—
9	C₂²×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	64	yes	direct product factor
13	C₂×#107	C₂²	1	48	yes	—
14	C₂×C₄×C₈	C₂²	1	12	yes	—
15	C₂²×Q₈	—	2	24	no	—
16	C₂²×Q₈	—	4	16	no	—
17	C₂³×C₄	—	2	4	no	—
18	C₂³×C₄	D₄	1	1	yes	characteristic
19	C₂³×C₄	D₄	1	1	yes	characteristic
20	C₂³×C₄	C₂³	1	1	yes	characteristic
21	C₂²×C₈	Q₈	1	16	yes	complemented
22	C₂²×C₈	C₂³	1	12	yes	—
23	C₄⋊Q₈	—	2	48	no	—
24	C₄⋊Q₈	C₂³	1	32	yes	—
25	C₂×C₄⋊C₄	—	2	24	no	—
26	C₂×C₄⋊C₄	—	2	12	no	—
27	C₂×C₄⋊C₄	C₂³	1	24	yes	—
28	C₂×C₄²	D₄	1	6	yes	—
29	C₂×C₄²	C₂³	1	6	yes	—
30	#107	C₂³	1	64	yes	—
31	C₄×C₈	C₂³	1	16	yes	—
32	C₂⁴	C₂×D₄	1	1	yes	characteristic
33	C₂×Q₈	—	2	32	no	—
34	C₂×Q₈	—	4	96	no	—
35	C₂²×C₄	—	2	24	no	—
36	C₂²×C₄	—	4	16	no	—
37	C₂²×C₄	D₈	1	8	yes	—
38	C₂²×C₄	Q₁₆	1	8	yes	—
39	C₂²×C₄	C₂×D₄	1	3	yes	—
40	C₂²×C₄	C₂×D₄	1	6	yes	—
41	C₂²×C₄	C₂×D₄	1	6	yes	—
42	C₂²×C₄	C₂×Q₈	1	8	yes	—
43	C₂²×C₄	C₂⁴	1	3	yes	—
44	C₂×C₈	C₂×Q₈	1	48	yes	complemented
45	C₂×C₈	C₂⁴	1	8	yes	—
46	C₄⋊C₄	—	2	32	no	—
47	C₄⋊C₄	—	2	24	no	—
48	C₄⋊C₄	C₂⁴	1	16	yes	—
49	C₄²	C₂×D₄	1	12	yes	—
50	C₄²	C₂⁴	1	4	yes	—
51	C₂³	C₄⋊Q₈	1	4	yes	—
52	C₂³	C₂×D₈	1	4	yes	—
53	C₂³	C₂×Q₁₆	1	4	yes	—
54	C₂³	C₂²×D₄	1	3	yes	—
55	Q₈	—	4	64	no	—
56	C₂×C₄	—	2	16	no	—
57	C₂×C₄	—	4	48	no	—
58	C₂×C₄	C₄⋊Q₈	1	12	yes	—
59	C₂×C₄	C₂×D₈	1	24	yes	—
60	C₂×C₄	C₂×Q₁₆	1	24	yes	—
61	C₂×C₄	C₂²×D₄	1	3	yes	—
62	C₂×C₄	C₂²×D₄	1	4	yes	—
63	C₂×C₄	C₂²×D₄	1	4	yes	—
64	C₂×C₄	C₂²×Q₈	1	12	yes	—
65	C₂×C₄	C₂⁵	1	1	yes	characteristic
66	C₈	C₂²×Q₈	1	16	yes	complemented
67	C₂²	#500	1	16	yes	direct product factor
68	C₂²	C₂×C₄⋊Q₈	1	6	yes	—
69	C₂²	C₂²×D₈	1	6	yes	—
70	C₂²	C₂²×Q₁₆	1	6	yes	—
71	C₂²	C₂³×D₄	1	1	yes	characteristic
72	C₄	—	4	16	no	—
73	C₄	C₂×C₄⋊Q₈	1	6	yes	—
74	C₄	C₂²×D₈	1	8	yes	—
75	C₄	C₂²×Q₁₆	1	8	yes	—
76	C₄	C₂³×Q₈	1	2	yes	—
77	C₂	C₂×#500	1	12	yes	direct product factor
78	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
79	C₂	C₂³×D₈	1	1	yes	characteristic
80	C₂	C₂³×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#500	C₂	direct (N × H)
12. #500	C₂²	direct (N × H)
21. C₂²×C₈	Q₈	semidirect
44. C₂×C₈	C₂×Q₈	semidirect
66. C₈	C₂²×Q₈	semidirect
67. C₂²	#500	direct (N × H)
77. C₂	C₂×#500	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊Q₈	2
C₂×#500	2
C₂²×#107	2
C₂²×C₄×C₈	2

Subgroups — C22×#500 #60654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#500 #60654